08:15-08:45   Sining Gong (Michigan State University)

High Order Unconditionally Strong Stability Preserving (SSP) Implicit Two-derivative Runge–Kutta Schemes and their Extensions

We construct a novel family of fully-implicit, high-order, unconditionally strong stability-preserving (SSP) second-derivative Runge–Kutta schemes designed for equations containing stiff terms. This approach has been validated to possess the asymptotic-preserving property, which preserves the asymptotic limit of the equation, and the scheme automatically reduces to a high order time discretization for the limiting system. Additionally, the inherent contractivity of the SSP method ensures the uniqueness of solutions in the stage equations of the implicit Runge-Kutta method. David I. Ketcheson has demonstrated the existence of a second-order unconditionally SSP Runge–Kutta method that incorporates downwind conditions. Our research, however, reveals that employing this method directly, even when integrating the backward derivative condition, fails to work. Instead, by leveraging this idea with the downwind conditions and integrating it with additive Runge–Kutta methods, we have developed a new family of second-order SSP methods that eliminate time-step limitations for various problems, including the Broadwell model, the radiative transport equation, and the Bhatnagar-Gross-Krook (BGK) kinetic equation. In this talk, we provide a summary of the key theorems and present the corresponding numerical results associated with the models. This is a joint work with Andrew Christlieb, Sigal Gottlieb and Zachary J. Grant.

08:45-09:15   David I. Ketcheson (King Abdullah University of Science & Technology)

Energy-Preserving Explicit Runge-Kutta Methods

 Using a recent characterization of energy-preserving B-series, we derive the explicit conditions on the coefficients of a Runge-Kutta method that ensure energy preservation up to a given order in the step size. Such pseudo-energy-preserving methods can be expected to behave like energy-preserving methods over moderate time intervals. We provide examples of pseudo-energy-preserving methods up to order six, and apply them to Hamiltonian ODE and PDE systems. We find that these methods exhibit significantly smaller errors, relative to other Runge-Kutta methods of the same order, for long-time simulations. 

09:15-09:45   Alexey Shevyakov (University of Saskatchewan)

Invariant Conservation Law-Preserving Wave Equation Discretizations with Applications to Fiber-Reinforced Materials

Symmetry- and conservation law-preserving finite difference discretizations are obtained for linear and nonlinear one-dimensional wave equations on five- and nine-point stencils, using the theory of Lie point symmetries of difference equations, and the discrete direct multiplier method of conservation law construction. In particular, for the linear wave equation, an explicit five-point scheme is presented that preserves the discrete analogs of its basic geometric point symmetries, and six of the corresponding conservation laws. For a class of nonlinear wave equations arising in hyperelasticity, a nine-point implicit scheme is constructed, preserving four point symmetries and three local conservation laws. Other discretization of the nonlinear wave equations preserving different subsets of conservation laws are discussed. 

04:00-04:30   Tri P. Nguyen, (University of California, Merced)

Simulating Strongly Magnetized Charged Particle Dynamics by Nyström Type Exponential Integration

The main component of particle-in-cell (PIC) methods widely employed in plasma physics simulations is solution of a particle pushing problem. It is the model of a charged particle moving under the influence of electromagnetic fields which has to be solved for very large numbers of particles in PIC codes. When the plasma is strongly magnetized, this problem is computationally challenging due numerical stiffness arising from the wide disparity in time scales between fast scale, highly oscillatory gyromotion and slow scale, macroscopic behavior of the system. We explored an alternative to numerical particle pushing using exponential integration techniques that solve linear problems exactly, are capable of yielding oscillatory solutions, and are A-stable. In particular, we developed Nyström-type exponential integrators that exploit the mathematical structure of the Newtonian equations of motion. Our numerical experiments show that these exponential integrators yield computational savings compared to such widely used fast schemes as Boris and Buneman. Moreover, while Boris is known to artificially enlarge gyroradius when stepping over the gyroperiod for linear problems, the new exponential schemes preserve gyroradius accurately even with step sizes far exceeding the gyroperiod. We discuss the advantages of the new exponential schemes for particle pushing problems and outline research directions for further possible improvements to the methods.

04:30-05:00   Samuel Van Fleet (University of Washington)

Fully Discrete Energy Conserving and Entropy Dissipative Particle Method for the Landau Equation 

At the semi-discrete level, the recently proposed particle method to the Landau equation by Carrillo et al. conserves mass, momentum, energy, and has a decaying entropy. When this method is coupled with the standard time discretization such as forward Euler, not all the above properties can be maintained. In this work, we show that using a properly designed discrete gradient method, conservation of mass, momentum and energy along with the decay of entropy are guaranteed at the fully discrete level. Several numerical examples are given to validate these results. 

05:00-05:30   Hannah Potgieter (Simon Fraser University)

Approximation of the First P-Laplace Eigenpair on Surfaces

The p-Laplace operator appears in variety of applications including image processing, optimal transport, and distance approximation. We approximate the first eigenpair of the p-Laplace operator with zero Dirichlet boundary conditions using a surface finite element method. In this talk we will discuss the p→∞ limit and its connection to the underlying geometry of our domain. Working with large p values presents numerical challenges which require careful treatment. We present computational results in 1D, planar domains, and surfaces lying in ℝ3.

05:30-06:00  Damien Tageddine (McGill University)

A Candidate Framework for Structure-Preserving Discretizations

In this talk, we present a possible general approach to structure-preserving discretizations and use it to study the approximation of differential structures defined by derivations. We determine the proper general setting in which one can expect convergence. It turns out that structure-preserving discretizations and their limits involving infinite dimensional operators should be looked through the lens of C∗-algebras. Indeed, it is only when a discretization is interpreted as a deformed structure from a C∗-algebra that one can understand the precise nature of the finite dimensional approximation and its limit as n→+∞. In fact, we show how the Berezin-Toeplitz quantization fits in naturally as a structure-preserving discretization of a differential algebra.